LMFDB - Embedded newform 275.4.b.c.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,4,Mod(199,275)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("275.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	275.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.2255252516$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	275.199
Dual form			275.4.b.c.199.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205i q^{2} -7.92820i q^{3} +0.535898 q^{4} -21.6603 q^{6} -3.07180i q^{7} -23.3205i q^{8} -35.8564 q^{9} +O(q^{10})$	\(q-2.73205i q^{2} -7.92820i q^{3} +0.535898 q^{4} -21.6603 q^{6} -3.07180i q^{7} -23.3205i q^{8} -35.8564 q^{9} -11.0000 q^{11} -4.24871i q^{12} +5.35898i q^{13} -8.39230 q^{14} -59.4256 q^{16} +41.2154i q^{17} +97.9615i q^{18} -139.923 q^{19} -24.3538 q^{21} +30.0526i q^{22} -111.354i q^{23} -184.890 q^{24} +14.6410 q^{26} +70.2154i q^{27} -1.64617i q^{28} +24.9948 q^{29} +31.4974 q^{31} -24.2102i q^{32} +87.2102i q^{33} +112.603 q^{34} -19.2154 q^{36} -13.1436i q^{37} +382.277i q^{38} +42.4871 q^{39} +261.072 q^{41} +66.5359i q^{42} -57.7128i q^{43} -5.89488 q^{44} -304.224 q^{46} +343.846i q^{47} +471.138i q^{48} +333.564 q^{49} +326.764 q^{51} +2.87187i q^{52} -342.995i q^{53} +191.832 q^{54} -71.6359 q^{56} +1109.34i q^{57} -68.2872i q^{58} -88.3693 q^{59} +738.697 q^{61} -86.0526i q^{62} +110.144i q^{63} -541.549 q^{64} +238.263 q^{66} -342.359i q^{67} +22.0873i q^{68} -882.836 q^{69} -207.364 q^{71} +836.190i q^{72} -1010.60i q^{73} -35.9090 q^{74} -74.9845 q^{76} +33.7898i q^{77} -116.077i q^{78} -1294.23 q^{79} -411.441 q^{81} -713.261i q^{82} +441.846i q^{83} -13.0512 q^{84} -157.674 q^{86} -198.164i q^{87} +256.526i q^{88} +1489.11 q^{89} +16.4617 q^{91} -59.6743i q^{92} -249.718i q^{93} +939.405 q^{94} -191.944 q^{96} -1346.42i q^{97} -911.314i q^{98} +394.420 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{4} - 52 q^{6} - 88 q^{9}+O(q^{10}) $ 4 * q + 16 * q^4 - 52 * q^6 - 88 * q^9	$ 4 q + 16 q^{4} - 52 q^{6} - 88 q^{9} - 44 q^{11} + 8 q^{14} - 16 q^{16} - 144 q^{19} + 152 q^{21} - 504 q^{24} - 80 q^{26} - 288 q^{29} - 68 q^{31} + 104 q^{34} - 160 q^{36} - 800 q^{39} + 1072 q^{41} - 176 q^{44} - 628 q^{46} + 780 q^{49} - 328 q^{51} + 220 q^{54} + 240 q^{56} - 1268 q^{59} + 1680 q^{61} - 448 q^{64} + 572 q^{66} - 1924 q^{69} - 1356 q^{71} - 12 q^{74} + 864 q^{76} - 632 q^{79} - 2588 q^{81} + 1472 q^{84} - 312 q^{86} + 3684 q^{89} + 2560 q^{91} + 1984 q^{94} - 1904 q^{96} + 968 q^{99}+O(q^{100}) $ 4 * q + 16 * q^4 - 52 * q^6 - 88 * q^9 - 44 * q^11 + 8 * q^14 - 16 * q^16 - 144 * q^19 + 152 * q^21 - 504 * q^24 - 80 * q^26 - 288 * q^29 - 68 * q^31 + 104 * q^34 - 160 * q^36 - 800 * q^39 + 1072 * q^41 - 176 * q^44 - 628 * q^46 + 780 * q^49 - 328 * q^51 + 220 * q^54 + 240 * q^56 - 1268 * q^59 + 1680 * q^61 - 448 * q^64 + 572 * q^66 - 1924 * q^69 - 1356 * q^71 - 12 * q^74 + 864 * q^76 - 632 * q^79 - 2588 * q^81 + 1472 * q^84 - 312 * q^86 + 3684 * q^89 + 2560 * q^91 + 1984 * q^94 - 1904 * q^96 + 968 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 2.73205i	− 0.965926i	−0.875641	−	0.482963i	$-0.839561\pi$
$2$	− 2.73205i	− 0.965926i	0.875641	−	0.482963i	$-0.160439\pi$
$3$	− 7.92820i	− 1.52578i	−0.646526	−	0.762892i	$-0.723779\pi$
$3$	− 7.92820i	− 1.52578i	0.646526	−	0.762892i	$-0.276221\pi$
$4$	0.535898	0.0669873
$5$	0	0
$5$	0	0
$6$	−21.6603	−1.47379
$7$	− 3.07180i	− 0.165861i	−0.996555	−	0.0829307i	$-0.973572\pi$
$7$	− 3.07180i	− 0.165861i	0.996555	−	0.0829307i	$-0.0264280\pi$
$8$	− 23.3205i	− 1.03063i
$9$	−35.8564	−1.32802
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	− 4.24871i	− 0.102208i
$13$	5.35898i	0.114332i	0.998365	+	0.0571659i	$0.0182064\pi$
$13$	5.35898i	0.114332i	−0.998365	+	0.0571659i	$0.981794\pi$
$14$	−8.39230	−0.160210
$15$	0	0
$16$	−59.4256	−0.928525
$17$	41.2154i	0.588012i	0.955804	+	0.294006i	$0.0949885\pi$
$17$	41.2154i	0.588012i	−0.955804	+	0.294006i	$0.905011\pi$
$18$	97.9615i	1.28276i
$19$	−139.923	−1.68950	−0.844751	−	0.535159i	$-0.820252\pi$
$19$	−139.923	−1.68950	−0.844751	+	0.535159i	$0.820252\pi$
$20$	0	0
$21$	−24.3538	−0.253069
$22$	30.0526i	0.291238i
$23$	− 111.354i	− 1.00952i	−0.863261	−	0.504758i	$-0.831582\pi$
$23$	− 111.354i	− 1.00952i	0.863261	−	0.504758i	$-0.168418\pi$
$24$	−184.890	−1.57252
$25$	0	0
$26$	14.6410	0.110436
$27$	70.2154i	0.500480i
$28$	− 1.64617i	− 0.0111106i
$29$	24.9948	0.160049	0.0800246	−	0.996793i	$-0.474500\pi$
$29$	24.9948	0.160049	0.0800246	+	0.996793i	$0.474500\pi$
$30$	0	0
$31$	31.4974	0.182487	0.0912436	−	0.995829i	$-0.470916\pi$
$31$	31.4974	0.182487	0.0912436	+	0.995829i	$0.470916\pi$
$32$	− 24.2102i	− 0.133744i
$33$	87.2102i	0.460041i
$34$	112.603	0.567976
$35$	0	0
$36$	−19.2154	−0.0889601
$37$	− 13.1436i	− 0.0583998i	−0.999574	−	0.0291999i	$-0.990704\pi$
$37$	− 13.1436i	− 0.0583998i	0.999574	−	0.0291999i	$-0.00929594\pi$
$38$	382.277i	1.63193i
$39$	42.4871	0.174446
$40$	0	0
$41$	261.072	0.994453	0.497226	−	0.867621i	$-0.334352\pi$
$41$	261.072	0.994453	0.497226	+	0.867621i	$0.334352\pi$
$42$	66.5359i	0.244446i
$43$	− 57.7128i	− 0.204677i	−0.994750	−	0.102339i	$-0.967367\pi$
$43$	− 57.7128i	− 0.204677i	0.994750	−	0.102339i	$-0.0326325\pi$
$44$	−5.89488	−0.0201974
$45$	0	0
$46$	−304.224	−0.975118
$47$	343.846i	1.06713i	0.845759	+	0.533565i	$0.179148\pi$
$47$	343.846i	1.06713i	−0.845759	+	0.533565i	$0.820852\pi$
$48$	471.138i	1.41673i
$49$	333.564	0.972490
$50$	0	0
$51$	326.764	0.897179
$52$	2.87187i	0.00765879i
$53$	− 342.995i	− 0.888943i	−0.895793	−	0.444471i	$-0.853392\pi$
$53$	− 342.995i	− 0.888943i	0.895793	−	0.444471i	$-0.146608\pi$
$54$	191.832	0.483426
$55$	0	0
$56$	−71.6359	−0.170942
$57$	1109.34i	2.57782i
$58$	− 68.2872i	− 0.154596i
$59$	−88.3693	−0.194995	−0.0974975	−	0.995236i	$-0.531084\pi$
$59$	−88.3693	−0.194995	−0.0974975	+	0.995236i	$0.531084\pi$
$60$	0	0
$61$	738.697	1.55050	0.775250	−	0.631654i	$-0.217624\pi$
$61$	738.697	1.55050	0.775250	+	0.631654i	$0.217624\pi$
$62$	− 86.0526i	− 0.176269i
$63$	110.144i	0.220266i
$64$	−541.549	−1.05771
$65$	0	0
$66$	238.263	0.444365
$67$	− 342.359i	− 0.624266i	−0.950038	−	0.312133i	$-0.898957\pi$
$67$	− 342.359i	− 0.624266i	0.950038	−	0.312133i	$-0.101043\pi$
$68$	22.0873i	0.0393893i
$69$	−882.836	−1.54030
$70$	0	0
$71$	−207.364	−0.346614	−0.173307	−	0.984868i	$-0.555445\pi$
$71$	−207.364	−0.346614	−0.173307	+	0.984868i	$0.555445\pi$
$72$	836.190i	1.36869i
$73$	− 1010.60i	− 1.62030i	−0.586224	−	0.810149i	$-0.699386\pi$
$73$	− 1010.60i	− 1.62030i	0.586224	−	0.810149i	$-0.300614\pi$
$74$	−35.9090	−0.0564099
$75$	0	0
$76$	−74.9845	−0.113175
$77$	33.7898i	0.0500091i
$78$	− 116.077i	− 0.168502i
$79$	−1294.23	−1.84319	−0.921593	−	0.388157i	$-0.873112\pi$
$79$	−1294.23	−1.84319	−0.921593	+	0.388157i	$0.873112\pi$
$80$	0	0
$81$	−411.441	−0.564391
$82$	− 713.261i	− 0.960568i
$83$	441.846i	0.584324i	0.956369	+	0.292162i	$0.0943747\pi$
$83$	441.846i	0.584324i	−0.956369	+	0.292162i	$0.905625\pi$
$84$	−13.0512	−0.0169524
$85$	0	0
$86$	−157.674	−0.197703
$87$	− 198.164i	− 0.244200i
$88$	256.526i	0.310747i
$89$	1489.11	1.77355	0.886773	−	0.462205i	$-0.152942\pi$
$89$	1489.11	1.77355	0.886773	+	0.462205i	$0.152942\pi$
$90$	0	0
$91$	16.4617	0.0189633
$92$	− 59.6743i	− 0.0676248i
$93$	− 249.718i	− 0.278436i
$94$	939.405	1.03077
$95$	0	0
$96$	−191.944	−0.204064
$97$	− 1346.42i	− 1.40936i	−0.709526	−	0.704679i	$-0.751091\pi$
$97$	− 1346.42i	− 1.40936i	0.709526	−	0.704679i	$-0.248909\pi$
$98$	− 911.314i	− 0.939353i
$99$	394.420	0.400412
$100$	0	0
$101$	−161.461	−0.159069	−0.0795347	−	0.996832i	$-0.525343\pi$
$101$	−161.461	−0.159069	−0.0795347	+	0.996832i	$0.525343\pi$
$102$	− 892.736i	− 0.866608i
$103$	− 34.7592i	− 0.0332517i	−0.999862	−	0.0166259i	$-0.994708\pi$
$103$	− 34.7592i	− 0.0332517i	0.999862	−	0.0166259i	$-0.00529242\pi$
$104$	124.974	0.117834
$105$	0	0
$106$	−937.079	−0.858653
$107$	− 832.179i	− 0.751867i	−0.926647	−	0.375934i	$-0.877322\pi$
$107$	− 832.179i	− 0.751867i	0.926647	−	0.375934i	$-0.122678\pi$
$108$	37.6283i	0.0335258i
$109$	−1044.26	−0.917629	−0.458815	−	0.888532i	$-0.651726\pi$
$109$	−1044.26	−0.917629	−0.458815	+	0.888532i	$0.651726\pi$
$110$	0	0
$111$	−104.205	−0.0891055
$112$	182.543i	0.154007i
$113$	295.082i	0.245654i	0.992428	+	0.122827i	$0.0391961\pi$
$113$	295.082i	0.245654i	−0.992428	+	0.122827i	$0.960804\pi$
$114$	3030.77	2.48998
$115$	0	0
$116$	13.3947	0.0107213
$117$	− 192.154i	− 0.151834i
$118$	241.429i	0.188351i
$119$	126.605	0.0975285
$120$	0	0
$121$	121.000	0.0909091
$122$	− 2018.16i	− 1.49767i
$123$	− 2069.83i	− 1.51732i
$124$	16.8794	0.0122243
$125$	0	0
$126$	300.918	0.212761
$127$	1317.60i	0.920618i	0.887759	+	0.460309i	$0.152261\pi$
$127$	1317.60i	0.920618i	−0.887759	+	0.460309i	$0.847739\pi$
$128$	1285.86i	0.887928i
$129$	−457.559	−0.312293
$130$	0	0
$131$	−1600.71	−1.06759	−0.533797	−	0.845612i	$-0.679235\pi$
$131$	−1600.71	−1.06759	−0.533797	+	0.845612i	$0.679235\pi$
$132$	46.7358i	0.0308169i
$133$	429.815i	0.280223i
$134$	−935.342	−0.602994
$135$	0	0
$136$	961.164	0.606023
$137$	− 1611.68i	− 1.00507i	−0.864556	−	0.502536i	$-0.832400\pi$
$137$	− 1611.68i	− 1.00507i	0.864556	−	0.502536i	$-0.167600\pi$
$138$	2411.95i	1.48782i
$139$	31.8619	0.0194424	0.00972120	−	0.999953i	$-0.496906\pi$
$139$	31.8619	0.0194424	0.00972120	+	0.999953i	$0.496906\pi$
$140$	0	0
$141$	2726.08	1.62821
$142$	566.529i	0.334803i
$143$	− 58.9488i	− 0.0344724i
$144$	2130.79	1.23310
$145$	0	0
$146$	−2761.01	−1.56509
$147$	− 2644.56i	− 1.48381i
$148$	− 7.04363i	− 0.00391205i
$149$	2428.34	1.33515	0.667576	−	0.744542i	$-0.267332\pi$
$149$	2428.34	1.33515	0.667576	+	0.744542i	$0.267332\pi$
$150$	0	0
$151$	−2576.68	−1.38866	−0.694328	−	0.719659i	$-0.744298\pi$
$151$	−2576.68	−1.38866	−0.694328	+	0.719659i	$0.744298\pi$
$152$	3263.08i	1.74125i
$153$	− 1477.84i	− 0.780889i
$154$	92.3154	0.0483051
$155$	0	0
$156$	22.7688	0.0116856
$157$	− 2475.94i	− 1.25861i	−0.777158	−	0.629305i	$-0.783340\pi$
$157$	− 2475.94i	− 1.25861i	0.777158	−	0.629305i	$-0.216660\pi$
$158$	3535.89i	1.78038i
$159$	−2719.33	−1.35633
$160$	0	0
$161$	−342.056	−0.167440
$162$	1124.08i	0.545160i
$163$	− 2725.11i	− 1.30949i	−0.755850	−	0.654745i	$-0.772776\pi$
$163$	− 2725.11i	− 1.30949i	0.755850	−	0.654745i	$-0.227224\pi$
$164$	139.908	0.0666157
$165$	0	0
$166$	1207.15	0.564414
$167$	− 2737.30i	− 1.26837i	−0.773180	−	0.634187i	$-0.781335\pi$
$167$	− 2737.30i	− 1.26837i	0.773180	−	0.634187i	$-0.218665\pi$
$168$	567.944i	0.260820i
$169$	2168.28	0.986928
$170$	0	0
$171$	5017.14	2.24368
$172$	− 30.9282i	− 0.0137108i
$173$	2307.42i	1.01404i	0.861933	+	0.507022i	$0.169254\pi$
$173$	2307.42i	1.01404i	−0.861933	+	0.507022i	$0.830746\pi$
$174$	−541.395	−0.235879
$175$	0	0
$176$	653.682	0.279961
$177$	700.610i	0.297520i
$178$	− 4068.33i	− 1.71311i
$179$	1312.15	0.547905	0.273953	−	0.961743i	$-0.411669\pi$
$179$	1312.15	0.547905	0.273953	+	0.961743i	$0.411669\pi$
$180$	0	0
$181$	−803.174	−0.329831	−0.164916	−	0.986308i	$-0.552735\pi$
$181$	−803.174	−0.329831	−0.164916	+	0.986308i	$0.552735\pi$
$182$	− 44.9742i	− 0.0183171i
$183$	− 5856.54i	− 2.36573i
$184$	−2596.83	−1.04044
$185$	0	0
$186$	−682.242	−0.268949
$187$	− 453.369i	− 0.177292i
$188$	184.267i	0.0714842i
$189$	215.687	0.0830103
$190$	0	0
$191$	1718.25	0.650932	0.325466	−	0.945554i	$-0.394479\pi$
$191$	1718.25	0.650932	0.325466	+	0.945554i	$0.394479\pi$
$192$	4293.51i	1.61384i
$193$	1340.18i	0.499837i	0.968267	+	0.249919i	$0.0804038\pi$
$193$	1340.18i	0.499837i	−0.968267	+	0.249919i	$0.919596\pi$
$194$	−3678.48	−1.36134
$195$	0	0
$196$	178.756	0.0651445
$197$	3518.33i	1.27244i	0.771508	+	0.636220i	$0.219503\pi$
$197$	3518.33i	1.27244i	−0.771508	+	0.636220i	$0.780497\pi$
$198$	− 1077.58i	− 0.386768i
$199$	−823.692	−0.293417	−0.146709	−	0.989180i	$-0.546868\pi$
$199$	−823.692	−0.293417	−0.146709	+	0.989180i	$0.546868\pi$
$200$	0	0
$201$	−2714.29	−0.952494
$202$	441.121i	0.153649i
$203$	− 76.7791i	− 0.0265460i
$204$	175.112	0.0600996
$205$	0	0
$206$	−94.9639	−0.0321187
$207$	3992.75i	1.34065i
$208$	− 318.461i	− 0.106160i
$209$	1539.15	0.509404
$210$	0	0
$211$	−107.343	−0.0350228	−0.0175114	−	0.999847i	$-0.505574\pi$
$211$	−107.343	−0.0350228	−0.0175114	+	0.999847i	$0.505574\pi$
$212$	− 183.810i	− 0.0595479i
$213$	1644.03i	0.528858i
$214$	−2273.56	−0.726248
$215$	0	0
$216$	1637.46	0.515810
$217$	− 96.7537i	− 0.0302676i
$218$	2852.96i	0.886362i
$219$	−8012.24	−2.47222
$220$	0	0
$221$	−220.873	−0.0672285
$222$	284.694i	0.0860693i
$223$	− 3933.68i	− 1.18125i	−0.806946	−	0.590625i	$-0.798881\pi$
$223$	− 3933.68i	− 1.18125i	0.806946	−	0.590625i	$-0.201119\pi$
$224$	−74.3689	−0.0221830
$225$	0	0
$226$	806.178	0.237284
$227$	1771.90i	0.518085i	0.965866	+	0.259042i	$0.0834069\pi$
$227$	1771.90i	0.518085i	−0.965866	+	0.259042i	$0.916593\pi$
$228$	594.493i	0.172681i
$229$	−1915.37	−0.552713	−0.276356	−	0.961055i	$-0.589127\pi$
$229$	−1915.37	−0.552713	−0.276356	+	0.961055i	$0.589127\pi$
$230$	0	0
$231$	267.892	0.0763031
$232$	− 582.892i	− 0.164952i
$233$	4396.32i	1.23610i	0.786137	+	0.618052i	$0.212078\pi$
$233$	4396.32i	1.23610i	−0.786137	+	0.618052i	$0.787922\pi$
$234$	−524.974	−0.146661
$235$	0	0
$236$	−47.3570	−0.0130622
$237$	10260.9i	2.81230i
$238$	− 345.892i	− 0.0942053i
$239$	4084.49	1.10546	0.552728	−	0.833362i	$-0.313587\pi$
$239$	4084.49	1.10546	0.552728	+	0.833362i	$0.313587\pi$
$240$	0	0
$241$	3908.58	1.04471	0.522353	−	0.852730i	$-0.325054\pi$
$241$	3908.58	1.04471	0.522353	+	0.852730i	$0.325054\pi$
$242$	− 330.578i	− 0.0878114i
$243$	5157.80i	1.36162i
$244$	395.867	0.103864
$245$	0	0
$246$	−5654.88	−1.46562
$247$	− 749.845i	− 0.193164i
$248$	− 734.536i	− 0.188077i
$249$	3503.05	0.891552
$250$	0	0
$251$	1094.89	0.275335	0.137667	−	0.990479i	$-0.456040\pi$
$251$	1094.89	0.275335	0.137667	+	0.990479i	$0.456040\pi$
$252$	59.0258i	0.0147551i
$253$	1224.89i	0.304381i
$254$	3599.76	0.889249
$255$	0	0
$256$	−819.364	−0.200040
$257$	− 783.179i	− 0.190091i	−0.995473	−	0.0950454i	$-0.969700\pi$
$257$	− 783.179i	− 0.190091i	0.995473	−	0.0950454i	$-0.0302996\pi$
$258$	1250.07i	0.301652i
$259$	−40.3744	−0.00968628
$260$	0	0
$261$	−896.225	−0.212548
$262$	4373.23i	1.03122i
$263$	6180.06i	1.44897i	0.689292	+	0.724484i	$0.257922\pi$
$263$	6180.06i	1.44897i	−0.689292	+	0.724484i	$0.742078\pi$
$264$	2033.79	0.474132
$265$	0	0
$266$	1174.28	0.270675
$267$	− 11806.0i	− 2.70605i
$268$	− 183.470i	− 0.0418179i
$269$	−986.965	−0.223704	−0.111852	−	0.993725i	$-0.535678\pi$
$269$	−986.965	−0.223704	−0.111852	+	0.993725i	$0.535678\pi$
$270$	0	0
$271$	4576.99	1.02595	0.512975	−	0.858404i	$-0.328543\pi$
$271$	4576.99	1.02595	0.512975	+	0.858404i	$0.328543\pi$
$272$	− 2449.25i	− 0.545984i
$273$	− 130.512i	− 0.0289338i
$274$	−4403.18	−0.970825
$275$	0	0
$276$	−473.110	−0.103181
$277$	− 567.836i	− 0.123169i	−0.998102	−	0.0615847i	$-0.980385\pi$
$277$	− 567.836i	− 0.123169i	0.998102	−	0.0615847i	$-0.0196154\pi$
$278$	− 87.0484i	− 0.0187799i
$279$	−1129.38	−0.242346
$280$	0	0
$281$	5311.01	1.12750	0.563752	−	0.825944i	$-0.309357\pi$
$281$	5311.01	1.12750	0.563752	+	0.825944i	$0.309357\pi$
$282$	− 7447.79i	− 1.57273i
$283$	− 4728.44i	− 0.993204i	−0.867978	−	0.496602i	$-0.834581\pi$
$283$	− 4728.44i	− 0.993204i	0.867978	−	0.496602i	$-0.165419\pi$
$284$	−111.126	−0.0232187
$285$	0	0
$286$	−161.051	−0.0332977
$287$	− 801.960i	− 0.164941i
$288$	868.092i	0.177614i
$289$	3214.29	0.654242
$290$	0	0
$291$	−10674.7	−2.15038
$292$	− 541.579i	− 0.108539i
$293$	2328.92i	0.464358i	0.972673	+	0.232179i	$0.0745854\pi$
$293$	2328.92i	0.464358i	−0.972673	+	0.232179i	$0.925415\pi$
$294$	−7225.08	−1.43325
$295$	0	0
$296$	−306.515	−0.0601886
$297$	− 772.369i	− 0.150900i
$298$	− 6634.36i	− 1.28966i
$299$	596.743	0.115420
$300$	0	0
$301$	−177.282	−0.0339481
$302$	7039.61i	1.34134i
$303$	1280.10i	0.242705i
$304$	8315.01	1.56875
$305$	0	0
$306$	−4037.52	−0.754280
$307$	1678.07i	0.311962i	0.987760	+	0.155981i	$0.0498539\pi$
$307$	1678.07i	0.311962i	−0.987760	+	0.155981i	$0.950146\pi$
$308$	18.1079i	0.00334997i
$309$	−275.578	−0.0507349
$310$	0	0
$311$	3572.71	0.651413	0.325707	−	0.945471i	$-0.394398\pi$
$311$	3572.71	0.651413	0.325707	+	0.945471i	$0.394398\pi$
$312$	− 990.821i	− 0.179789i
$313$	7184.36i	1.29739i	0.761047	+	0.648697i	$0.224686\pi$
$313$	7184.36i	1.29739i	−0.761047	+	0.648697i	$0.775314\pi$
$314$	−6764.40	−1.21572
$315$	0	0
$316$	−693.573	−0.123470
$317$	15.7077i	0.00278306i	0.999999	+	0.00139153i	$0.000442938\pi$
$317$	15.7077i	0.00278306i	−0.999999	+	0.00139153i	$0.999557\pi$
$318$	7429.36i	1.31012i
$319$	−274.943	−0.0482566
$320$	0	0
$321$	−6597.69	−1.14719
$322$	934.515i	0.161734i
$323$	− 5766.98i	− 0.993447i
$324$	−220.491	−0.0378070
$325$	0	0
$326$	−7445.13	−1.26487
$327$	8279.08i	1.40010i
$328$	− 6088.33i	− 1.02491i
$329$	1056.23	0.176996
$330$	0	0
$331$	−1318.95	−0.219022	−0.109511	−	0.993986i	$-0.534928\pi$
$331$	−1318.95	−0.219022	−0.109511	+	0.993986i	$0.534928\pi$
$332$	236.785i	0.0391423i
$333$	471.282i	0.0775558i
$334$	−7478.43	−1.22515
$335$	0	0
$336$	1447.24	0.234981
$337$	239.183i	0.0386621i	0.999813	+	0.0193310i	$0.00615365\pi$
$337$	239.183i	0.0386621i	−0.999813	+	0.0193310i	$0.993846\pi$
$338$	− 5923.85i	− 0.953299i
$339$	2339.47	0.374816
$340$	0	0
$341$	−346.472	−0.0550220
$342$	− 13707.1i	− 2.16723i
$343$	− 2078.27i	− 0.327160i
$344$	−1345.89	−0.210947
$345$	0	0
$346$	6303.98	0.979491
$347$	5862.79i	0.907006i	0.891255	+	0.453503i	$0.149826\pi$
$347$	5862.79i	0.907006i	−0.891255	+	0.453503i	$0.850174\pi$
$348$	− 106.196i	− 0.0163583i
$349$	−3491.73	−0.535553	−0.267776	−	0.963481i	$-0.586289\pi$
$349$	−3491.73	−0.535553	−0.267776	+	0.963481i	$0.586289\pi$
$350$	0	0
$351$	−376.283	−0.0572208
$352$	266.313i	0.0403253i
$353$	− 10916.7i	− 1.64600i	−0.568043	−	0.822999i	$-0.692299\pi$
$353$	− 10916.7i	− 1.64600i	0.568043	−	0.822999i	$-0.307701\pi$
$354$	1914.10	0.287382
$355$	0	0
$356$	798.013	0.118805
$357$	− 1003.75i	− 0.148807i
$358$	− 3584.87i	− 0.529236i
$359$	11500.7	1.69077	0.845384	−	0.534160i	$-0.179372\pi$
$359$	11500.7	1.69077	0.845384	+	0.534160i	$0.179372\pi$
$360$	0	0
$361$	12719.5	1.85442
$362$	2194.31i	0.318592i
$363$	− 959.313i	− 0.138708i
$364$	8.82180	0.00127030
$365$	0	0
$366$	−16000.4	−2.28512
$367$	− 6767.01i	− 0.962493i	−0.876585	−	0.481246i	$-0.840184\pi$
$367$	− 6767.01i	− 0.962493i	0.876585	−	0.481246i	$-0.159816\pi$
$368$	6617.27i	0.937362i
$369$	−9361.10	−1.32065
$370$	0	0
$371$	−1053.61	−0.147441
$372$	− 133.823i	− 0.0186517i
$373$	− 5310.22i	− 0.737139i	−0.929600	−	0.368569i	$-0.879848\pi$
$373$	− 5310.22i	− 0.737139i	0.929600	−	0.368569i	$-0.120152\pi$
$374$	−1238.63	−0.171251
$375$	0	0
$376$	8018.67	1.09982
$377$	133.947i	0.0182987i
$378$	− 589.269i	− 0.0801818i
$379$	838.267	0.113612	0.0568059	−	0.998385i	$-0.481908\pi$
$379$	838.267	0.113612	0.0568059	+	0.998385i	$0.481908\pi$
$380$	0	0
$381$	10446.2	1.40466
$382$	− 4694.34i	− 0.628752i
$383$	− 2832.16i	− 0.377851i	−0.981991	−	0.188925i	$-0.939500\pi$
$383$	− 2832.16i	− 0.377851i	0.981991	−	0.188925i	$-0.0605004\pi$
$384$	10194.5	1.35479
$385$	0	0
$386$	3661.45	0.482806
$387$	2069.37i	0.271814i
$388$	− 721.542i	− 0.0944091i
$389$	−3111.25	−0.405519	−0.202759	−	0.979229i	$-0.564991\pi$
$389$	−3111.25	−0.405519	−0.202759	+	0.979229i	$0.564991\pi$
$390$	0	0
$391$	4589.49	0.593608
$392$	− 7778.88i	− 1.00228i
$393$	12690.8i	1.62892i
$394$	9612.25	1.22908
$395$	0	0
$396$	211.369	0.0268225
$397$	− 14208.7i	− 1.79626i	−0.439728	−	0.898131i	$-0.644925\pi$
$397$	− 14208.7i	− 1.79626i	0.439728	−	0.898131i	$-0.355075\pi$
$398$	2250.37i	0.283419i
$399$	3407.66	0.427560
$400$	0	0
$401$	−6261.68	−0.779784	−0.389892	−	0.920861i	$-0.627488\pi$
$401$	−6261.68	−0.779784	−0.389892	+	0.920861i	$0.627488\pi$
$402$	7415.58i	0.920039i
$403$	168.794i	0.0208641i
$404$	−86.5269	−0.0106556
$405$	0	0
$406$	−209.764	−0.0256415
$407$	144.580i	0.0176082i
$408$	− 7620.30i	− 0.924660i
$409$	4192.50	0.506860	0.253430	−	0.967354i	$-0.418441\pi$
$409$	4192.50	0.506860	0.253430	+	0.967354i	$0.418441\pi$
$410$	0	0
$411$	−12777.7	−1.53352
$412$	− 18.6274i	− 0.00222744i
$413$	271.453i	0.0323421i
$414$	10908.4	1.29497
$415$	0	0
$416$	129.742	0.0152912
$417$	− 252.608i	− 0.0296649i
$418$	− 4205.05i	− 0.492047i
$419$	9287.15	1.08283	0.541416	−	0.840755i	$-0.317888\pi$
$419$	9287.15	1.08283	0.541416	+	0.840755i	$0.317888\pi$
$420$	0	0
$421$	13146.0	1.52185	0.760923	−	0.648842i	$-0.224746\pi$
$421$	13146.0	1.52185	0.760923	+	0.648842i	$0.224746\pi$
$422$	293.267i	0.0338294i
$423$	− 12329.1i	− 1.41716i
$424$	−7998.81	−0.916172
$425$	0	0
$426$	4491.56	0.510838
$427$	− 2269.13i	− 0.257168i
$428$	− 445.964i	− 0.0503656i
$429$	−467.358	−0.0525974
$430$	0	0
$431$	4909.67	0.548701	0.274351	−	0.961630i	$-0.411537\pi$
$431$	4909.67	0.548701	0.274351	+	0.961630i	$0.411537\pi$
$432$	− 4172.59i	− 0.464708i
$433$	− 11743.3i	− 1.30334i	−0.758502	−	0.651671i	$-0.774068\pi$
$433$	− 11743.3i	− 1.30334i	0.758502	−	0.651671i	$-0.225932\pi$
$434$	−264.336	−0.0292363
$435$	0	0
$436$	−559.615	−0.0614695
$437$	15581.0i	1.70558i
$438$	21889.8i	2.38798i
$439$	11824.2	1.28551	0.642754	−	0.766073i	$-0.277792\pi$
$439$	11824.2	1.28551	0.642754	+	0.766073i	$0.277792\pi$
$440$	0	0
$441$	−11960.4	−1.29148
$442$	603.435i	0.0649377i
$443$	10102.1i	1.08344i	0.840558	+	0.541722i	$0.182228\pi$
$443$	10102.1i	1.08344i	−0.840558	+	0.541722i	$0.817772\pi$
$444$	−55.8433	−0.00596894
$445$	0	0
$446$	−10747.0	−1.14100
$447$	− 19252.4i	− 2.03715i
$448$	1663.53i	0.175434i
$449$	345.254	0.0362885	0.0181443	−	0.999835i	$-0.494224\pi$
$449$	345.254	0.0362885	0.0181443	+	0.999835i	$0.494224\pi$
$450$	0	0
$451$	−2871.79	−0.299839
$452$	158.134i	0.0164557i
$453$	20428.4i	2.11879i
$454$	4840.93	0.500431
$455$	0	0
$456$	25870.3	2.65678
$457$	10567.1i	1.08164i	0.841138	+	0.540821i	$0.181886\pi$
$457$	10567.1i	1.08164i	−0.841138	+	0.540821i	$0.818114\pi$
$458$	5232.89i	0.533879i
$459$	−2893.95	−0.294288
$460$	0	0
$461$	4733.96	0.478270	0.239135	−	0.970986i	$-0.423136\pi$
$461$	4733.96	0.478270	0.239135	+	0.970986i	$0.423136\pi$
$462$	− 731.895i	− 0.0737031i
$463$	3431.20i	0.344409i	0.985061	+	0.172204i	$0.0550890\pi$
$463$	3431.20i	0.344409i	−0.985061	+	0.172204i	$0.944911\pi$
$464$	−1485.33	−0.148610
$465$	0	0
$466$	12011.0	1.19399
$467$	− 5116.96i	− 0.507034i	−0.967331	−	0.253517i	$-0.918413\pi$
$467$	− 5116.96i	− 0.507034i	0.967331	−	0.253517i	$-0.0815873\pi$
$468$	− 102.975i	− 0.0101710i
$469$	−1051.66	−0.103542
$470$	0	0
$471$	−19629.8	−1.92037
$472$	2060.82i	0.200968i
$473$	634.841i	0.0617125i
$474$	28033.2	2.71648
$475$	0	0
$476$	67.8476	0.00653317
$477$	12298.6i	1.18053i
$478$	− 11159.0i	− 1.06779i
$479$	−11566.9	−1.10335	−0.551675	−	0.834059i	$-0.686011\pi$
$479$	−11566.9	−1.10335	−0.551675	+	0.834059i	$0.686011\pi$
$480$	0	0
$481$	70.4363	0.00667696
$482$	− 10678.4i	− 1.00911i
$483$	2711.89i	0.255477i
$484$	64.8437	0.00608975
$485$	0	0
$486$	14091.4	1.31522
$487$	18326.5i	1.70525i	0.522527	+	0.852623i	$0.324990\pi$
$487$	18326.5i	1.70525i	−0.522527	+	0.852623i	$0.675010\pi$
$488$	− 17226.8i	− 1.59799i
$489$	−21605.2	−1.99800
$490$	0	0
$491$	−7617.58	−0.700156	−0.350078	−	0.936721i	$-0.613845\pi$
$491$	−7617.58	−0.700156	−0.350078	+	0.936721i	$0.613845\pi$
$492$	− 1109.22i	− 0.101641i
$493$	1030.17i	0.0941108i
$494$	−2048.62	−0.186582
$495$	0	0
$496$	−1871.75	−0.169444
$497$	636.980i	0.0574899i
$498$	− 9570.50i	− 0.861173i
$499$	−12909.1	−1.15810	−0.579050	−	0.815292i	$-0.696576\pi$
$499$	−12909.1	−1.15810	−0.579050	+	0.815292i	$0.696576\pi$
$500$	0	0
$501$	−21701.8	−1.93526
$502$	− 2991.30i	− 0.265953i
$503$	10165.7i	0.901121i	0.892746	+	0.450561i	$0.148776\pi$
$503$	10165.7i	0.901121i	−0.892746	+	0.450561i	$0.851224\pi$
$504$	2568.60	0.227013
$505$	0	0
$506$	3346.47	0.294009
$507$	− 17190.6i	− 1.50584i
$508$	706.102i	0.0616697i
$509$	−6449.93	−0.561666	−0.280833	−	0.959757i	$-0.590611\pi$
$509$	−6449.93	−0.561666	−0.280833	+	0.959757i	$0.590611\pi$
$510$	0	0
$511$	−3104.36	−0.268745
$512$	12525.4i	1.08115i
$513$	− 9824.75i	− 0.845562i
$514$	−2139.68	−0.183614
$515$	0	0
$516$	−245.205	−0.0209197
$517$	− 3782.31i	− 0.321752i
$518$	110.305i	0.00935623i
$519$	18293.7	1.54721
$520$	0	0
$521$	−19327.4	−1.62524	−0.812620	−	0.582794i	$-0.801959\pi$
$521$	−19327.4	−1.62524	−0.812620	+	0.582794i	$0.801959\pi$
$522$	2448.53i	0.205305i
$523$	6259.09i	0.523310i	0.965161	+	0.261655i	$0.0842682\pi$
$523$	6259.09i	0.523310i	−0.965161	+	0.261655i	$0.915732\pi$
$524$	−857.819	−0.0715153
$525$	0	0
$526$	16884.2	1.39960
$527$	1298.18i	0.107305i
$528$	− 5182.52i	− 0.427160i
$529$	−232.675	−0.0191235
$530$	0	0
$531$	3168.61	0.258956
$532$	230.337i	0.0187714i
$533$	1399.08i	0.113698i
$534$	−32254.6	−2.61384
$535$	0	0
$536$	−7983.99	−0.643387
$537$	− 10403.0i	− 0.835985i
$538$	2696.44i	0.216081i
$539$	−3669.20	−0.293217
$540$	0	0
$541$	−14008.2	−1.11323	−0.556616	−	0.830770i	$-0.687900\pi$
$541$	−14008.2	−1.11323	−0.556616	+	0.830770i	$0.687900\pi$
$542$	− 12504.6i	− 0.990991i
$543$	6367.72i	0.503251i
$544$	997.834	0.0786430
$545$	0	0
$546$	−356.565	−0.0279479
$547$	4949.45i	0.386879i	0.981112	+	0.193440i	$0.0619644\pi$
$547$	4949.45i	0.386879i	−0.981112	+	0.193440i	$0.938036\pi$
$548$	− 863.695i	− 0.0673270i
$549$	−26487.0	−2.05909
$550$	0	0
$551$	−3497.35	−0.270404
$552$	20588.2i	1.58748i
$553$	3975.60i	0.305714i
$554$	−1551.36	−0.118973
$555$	0	0
$556$	17.0748	0.00130239
$557$	3801.58i	0.289188i	0.989491	+	0.144594i	$0.0461877\pi$
$557$	3801.58i	0.289188i	−0.989491	+	0.144594i	$0.953812\pi$
$558$	3085.54i	0.234088i
$559$	309.282	0.0234011
$560$	0	0
$561$	−3594.40	−0.270510
$562$	− 14510.0i	− 1.08908i
$563$	− 9900.11i	− 0.741101i	−0.928812	−	0.370551i	$-0.879169\pi$
$563$	− 9900.11i	− 0.741101i	0.928812	−	0.370551i	$-0.120831\pi$
$564$	1460.90	0.109069
$565$	0	0
$566$	−12918.3	−0.959361
$567$	1263.86i	0.0936107i
$568$	4835.84i	0.357231i
$569$	−5329.16	−0.392636	−0.196318	−	0.980540i	$-0.562898\pi$
$569$	−5329.16	−0.392636	−0.196318	+	0.980540i	$0.562898\pi$
$570$	0	0
$571$	−16962.6	−1.24319	−0.621597	−	0.783337i	$-0.713516\pi$
$571$	−16962.6	−1.24319	−0.621597	+	0.783337i	$0.713516\pi$
$572$	− 31.5906i	− 0.00230921i
$573$	− 13622.6i	− 0.993181i
$574$	−2190.99	−0.159321
$575$	0	0
$576$	19418.0	1.40466
$577$	15487.0i	1.11738i	0.829375	+	0.558692i	$0.188697\pi$
$577$	15487.0i	1.11738i	−0.829375	+	0.558692i	$0.811303\pi$
$578$	− 8781.61i	− 0.631949i
$579$	10625.3	0.762643
$580$	0	0
$581$	1357.26	0.0969169
$582$	29163.7i	2.07710i
$583$	3772.94i	0.268026i
$584$	−23567.7	−1.66993
$585$	0	0
$586$	6362.72	0.448535
$587$	− 11084.2i	− 0.779373i	−0.920948	−	0.389686i	$-0.872583\pi$
$587$	− 11084.2i	− 0.779373i	0.920948	−	0.389686i	$-0.127417\pi$
$588$	− 1417.22i	− 0.0993964i
$589$	−4407.22	−0.308313
$590$	0	0
$591$	27894.0	1.94147
$592$	781.066i	0.0542257i
$593$	4349.68i	0.301214i	0.988594	+	0.150607i	$0.0481228\pi$
$593$	4349.68i	0.301214i	−0.988594	+	0.150607i	$0.951877\pi$
$594$	−2110.15	−0.145759
$595$	0	0
$596$	1301.34	0.0894382
$597$	6530.40i	0.447691i
$598$	− 1630.33i	− 0.111487i
$599$	−13183.9	−0.899299	−0.449650	−	0.893205i	$-0.648451\pi$
$599$	−13183.9	−0.899299	−0.449650	+	0.893205i	$0.648451\pi$
$600$	0	0
$601$	−18765.0	−1.27361	−0.636806	−	0.771024i	$-0.719745\pi$
$601$	−18765.0	−1.27361	−0.636806	+	0.771024i	$0.719745\pi$
$602$	484.344i	0.0327913i
$603$	12275.8i	0.829034i
$604$	−1380.84	−0.0930223
$605$	0	0
$606$	3497.29	0.234435
$607$	− 21871.4i	− 1.46249i	−0.682116	−	0.731244i	$-0.738940\pi$
$607$	− 21871.4i	− 1.46249i	0.682116	−	0.731244i	$-0.261060\pi$
$608$	3387.57i	0.225961i
$609$	−608.720	−0.0405034
$610$	0	0
$611$	−1842.67	−0.122007
$612$	− 791.970i	− 0.0523096i
$613$	− 3527.85i	− 0.232445i	−0.993223	−	0.116222i	$-0.962921\pi$
$613$	− 3527.85i	− 0.232445i	0.993223	−	0.116222i	$-0.0370785\pi$
$614$	4584.56	0.301332
$615$	0	0
$616$	787.994	0.0515409
$617$	22728.1i	1.48298i	0.670963	+	0.741490i	$0.265881\pi$
$617$	22728.1i	1.48298i	−0.670963	+	0.741490i	$0.734119\pi$
$618$	752.893i	0.0490062i
$619$	21443.3	1.39237	0.696187	−	0.717861i	$-0.254879\pi$
$619$	21443.3	1.39237	0.696187	+	0.717861i	$0.254879\pi$
$620$	0	0
$621$	7818.75	0.505243
$622$	− 9760.81i	− 0.629217i
$623$	− 4574.25i	− 0.294163i
$624$	−2524.82	−0.161977
$625$	0	0
$626$	19628.0	1.25319
$627$	− 12202.7i	− 0.777240i
$628$	− 1326.85i	− 0.0843109i
$629$	541.718	0.0343398
$630$	0	0
$631$	21532.0	1.35844	0.679219	−	0.733936i	$-0.262319\pi$
$631$	21532.0	1.35844	0.679219	+	0.733936i	$0.262319\pi$
$632$	30182.0i	1.89964i
$633$	851.038i	0.0534372i
$634$	42.9141	0.00268823
$635$	0	0
$636$	−1457.29	−0.0908572
$637$	1787.56i	0.111187i
$638$	751.159i	0.0466123i
$639$	7435.33	0.460309
$640$	0	0
$641$	20148.3	1.24151	0.620756	−	0.784004i	$-0.286826\pi$
$641$	20148.3	1.24151	0.620756	+	0.784004i	$0.286826\pi$
$642$	18025.2i	1.10810i
$643$	28869.7i	1.77062i	0.465000	+	0.885310i	$0.346054\pi$
$643$	28869.7i	1.77062i	−0.465000	+	0.885310i	$0.653946\pi$
$644$	−183.307	−0.0112163
$645$	0	0
$646$	−15755.7	−0.959597
$647$	1590.02i	0.0966155i	0.998833	+	0.0483077i	$0.0153828\pi$
$647$	1590.02i	0.0966155i	−0.998833	+	0.0483077i	$0.984617\pi$
$648$	9595.02i	0.581679i
$649$	972.062	0.0587932
$650$	0	0
$651$	−767.083	−0.0461818
$652$	− 1460.38i	− 0.0877192i
$653$	20028.1i	1.20024i	0.799909	+	0.600122i	$0.204881\pi$
$653$	20028.1i	1.20024i	−0.799909	+	0.600122i	$0.795119\pi$
$654$	22618.9	1.35240
$655$	0	0
$656$	−15514.4	−0.923375
$657$	36236.5i	2.15178i
$658$	− 2885.66i	− 0.170965i
$659$	10520.7	0.621897	0.310948	−	0.950427i	$-0.399353\pi$
$659$	10520.7	0.621897	0.310948	+	0.950427i	$0.399353\pi$
$660$	0	0
$661$	3295.83	0.193938	0.0969690	−	0.995287i	$-0.469085\pi$
$661$	3295.83	0.193938	0.0969690	+	0.995287i	$0.469085\pi$
$662$	3603.45i	0.211559i
$663$	1751.12i	0.102576i
$664$	10304.1	0.602222
$665$	0	0
$666$	1287.57	0.0749132
$667$	− 2783.27i	− 0.161572i
$668$	− 1466.91i	− 0.0849649i
$669$	−31187.0	−1.80233
$670$	0	0
$671$	−8125.67	−0.467493
$672$	589.612i	0.0338464i
$673$	− 1187.64i	− 0.0680239i	−0.999421	−	0.0340119i	$-0.989172\pi$
$673$	− 1187.64i	− 0.0680239i	0.999421	−	0.0340119i	$-0.0108284\pi$
$674$	653.460	0.0373447
$675$	0	0
$676$	1161.98	0.0661117
$677$	− 13221.4i	− 0.750574i	−0.926909	−	0.375287i	$-0.877544\pi$
$677$	− 13221.4i	− 0.750574i	0.926909	−	0.375287i	$-0.122456\pi$
$678$	− 6391.55i	− 0.362044i
$679$	−4135.91	−0.233758
$680$	0	0
$681$	14048.0	0.790485
$682$	946.578i	0.0531471i
$683$	− 13831.4i	− 0.774882i	−0.921894	−	0.387441i	$-0.873359\pi$
$683$	− 13831.4i	− 0.774882i	0.921894	−	0.387441i	$-0.126641\pi$
$684$	2688.68	0.150298
$685$	0	0
$686$	−5677.93	−0.316012
$687$	15185.4i	0.843320i
$688$	3429.62i	0.190048i
$689$	1838.10	0.101635
$690$	0	0
$691$	−9817.07	−0.540462	−0.270231	−	0.962796i	$-0.587100\pi$
$691$	−9817.07	−0.540462	−0.270231	+	0.962796i	$0.587100\pi$
$692$	1236.54i	0.0679280i
$693$	− 1211.58i	− 0.0664128i
$694$	16017.4	0.876101
$695$	0	0
$696$	−4621.29	−0.251680
$697$	10760.2i	0.584750i
$698$	9539.58i	0.517304i
$699$	34854.9	1.88603
$700$	0	0
$701$	29949.8	1.61368	0.806838	−	0.590773i	$-0.201177\pi$
$701$	29949.8	1.61368	0.806838	+	0.590773i	$0.201177\pi$
$702$	1028.02i	0.0552711i
$703$	1839.09i	0.0986667i
$704$	5957.03	0.318912
$705$	0	0
$706$	−29825.0	−1.58991
$707$	495.976i	0.0263835i
$708$	375.456i	0.0199301i
$709$	−11307.5	−0.598959	−0.299479	−	0.954103i	$-0.596813\pi$
$709$	−11307.5	−0.598959	−0.299479	+	0.954103i	$0.596813\pi$
$710$	0	0
$711$	46406.3	2.44778
$712$	− 34726.9i	− 1.82787i
$713$	− 3507.36i	− 0.184224i
$714$	−2742.30	−0.143737
$715$	0	0
$716$	703.181	0.0367027
$717$	− 32382.7i	− 1.68669i
$718$	− 31420.6i	− 1.63316i
$719$	32623.4	1.69214	0.846070	−	0.533071i	$-0.178962\pi$
$719$	32623.4	1.69214	0.846070	+	0.533071i	$0.178962\pi$
$720$	0	0
$721$	−106.773	−0.00551518
$722$	− 34750.2i	− 1.79123i
$723$	− 30988.0i	− 1.59399i
$724$	−430.420	−0.0220945
$725$	0	0
$726$	−2620.89	−0.133981
$727$	502.545i	0.0256373i	0.999918	+	0.0128187i	$0.00408042\pi$
$727$	502.545i	0.0256373i	−0.999918	+	0.0128187i	$0.995920\pi$
$728$	− 383.895i	− 0.0195441i
$729$	29783.2	1.51314
$730$	0	0
$731$	2378.66	0.120353
$732$	− 3138.51i	− 0.158474i
$733$	8631.37i	0.434935i	0.976068	+	0.217467i	$0.0697796\pi$
$733$	8631.37i	0.434935i	−0.976068	+	0.217467i	$0.930220\pi$
$734$	−18487.8	−0.929697
$735$	0	0
$736$	−2695.90	−0.135017
$737$	3765.95i	0.188223i
$738$	25575.0i	1.27565i
$739$	18357.5	0.913792	0.456896	−	0.889520i	$-0.348961\pi$
$739$	18357.5	0.913792	0.456896	+	0.889520i	$0.348961\pi$
$740$	0	0
$741$	−5944.93	−0.294726
$742$	2878.52i	0.142417i
$743$	11182.6i	0.552155i	0.961135	+	0.276078i	$0.0890347\pi$
$743$	11182.6i	0.552155i	−0.961135	+	0.276078i	$0.910965\pi$
$744$	−5823.55	−0.286965
$745$	0	0
$746$	−14507.8	−0.712021
$747$	− 15843.0i	− 0.775991i
$748$	− 242.960i	− 0.0118763i
$749$	−2556.29	−0.124706
$750$	0	0
$751$	16733.4	0.813063	0.406531	−	0.913637i	$-0.366738\pi$
$751$	16733.4	0.813063	0.406531	+	0.913637i	$0.366738\pi$
$752$	− 20433.3i	− 0.990857i
$753$	− 8680.53i	− 0.420101i
$754$	365.950	0.0176752
$755$	0	0
$756$	115.587	0.00556064
$757$	24402.4i	1.17163i	0.810446	+	0.585813i	$0.199225\pi$
$757$	24402.4i	1.17163i	−0.810446	+	0.585813i	$0.800775\pi$
$758$	− 2290.19i	− 0.109741i
$759$	9711.19	0.464419
$760$	0	0
$761$	8469.33	0.403434	0.201717	−	0.979444i	$-0.435348\pi$
$761$	8469.33	0.403434	0.201717	+	0.979444i	$0.435348\pi$
$762$	− 28539.7i	− 1.35680i
$763$	3207.74i	0.152199i
$764$	920.805	0.0436042
$765$	0	0
$766$	−7737.62	−0.364976
$767$	− 473.570i	− 0.0222941i
$768$	6496.08i	0.305218i
$769$	−32834.7	−1.53973	−0.769864	−	0.638208i	$-0.779676\pi$
$769$	−32834.7	−1.53973	−0.769864	+	0.638208i	$0.779676\pi$
$770$	0	0
$771$	−6209.20	−0.290038
$772$	718.202i	0.0334827i
$773$	− 35571.4i	− 1.65513i	−0.561371	−	0.827564i	$-0.689726\pi$
$773$	− 35571.4i	− 1.65513i	0.561371	−	0.827564i	$-0.310274\pi$
$774$	5653.64	0.262553
$775$	0	0
$776$	−31399.1	−1.45253
$777$	320.097i	0.0147792i
$778$	8500.11i	0.391701i
$779$	−36530.0	−1.68013
$780$	0	0
$781$	2281.01	0.104508
$782$	− 12538.7i	− 0.573381i
$783$	1755.02i	0.0801014i
$784$	−19822.3	−0.902982
$785$	0	0
$786$	34671.8	1.57341
$787$	− 15729.6i	− 0.712452i	−0.934400	−	0.356226i	$-0.884063\pi$
$787$	− 15729.6i	− 0.712452i	0.934400	−	0.356226i	$-0.115937\pi$
$788$	1885.47i	0.0852373i
$789$	48996.7	2.21081
$790$	0	0
$791$	906.431	0.0407446
$792$	− 9198.09i	− 0.412676i
$793$	3958.67i	0.177272i
$794$	−38819.0	−1.73506
$795$	0	0
$796$	−441.415	−0.0196552
$797$	− 7888.07i	− 0.350577i	−0.984517	−	0.175288i	$-0.943914\pi$
$797$	− 7888.07i	− 0.350577i	0.984517	−	0.175288i	$-0.0560858\pi$
$798$	− 9309.91i	− 0.412991i
$799$	−14171.8	−0.627485
$800$	0	0
$801$	−53394.2	−2.35530
$802$	17107.2i	0.753214i
$803$	11116.6i	0.488538i
$804$	−1454.58	−0.0638050
$805$	0	0
$806$	461.154	0.0201532
$807$	7824.86i	0.341323i
$808$	3765.36i	0.163942i
$809$	−5896.97	−0.256275	−0.128138	−	0.991756i	$-0.540900\pi$
$809$	−5896.97	−0.256275	−0.128138	+	0.991756i	$0.540900\pi$
$810$	0	0
$811$	14197.9	0.614744	0.307372	−	0.951589i	$-0.400550\pi$
$811$	14197.9	0.614744	0.307372	+	0.951589i	$0.400550\pi$
$812$	− 41.1458i	− 0.00177824i
$813$	− 36287.3i	− 1.56538i
$814$	394.999	0.0170082
$815$	0	0
$816$	−19418.2	−0.833053
$817$	8075.35i	0.345803i
$818$	− 11454.1i	− 0.489589i
$819$	−590.258	−0.0251835
$820$	0	0
$821$	−19841.7	−0.843459	−0.421729	−	0.906722i	$-0.638577\pi$
$821$	−19841.7	−0.843459	−0.421729	+	0.906722i	$0.638577\pi$
$822$	34909.3i	1.48127i
$823$	− 28202.2i	− 1.19449i	−0.802058	−	0.597246i	$-0.796262\pi$
$823$	− 28202.2i	− 1.19449i	0.802058	−	0.597246i	$-0.203738\pi$
$824$	−810.602	−0.0342702
$825$	0	0
$826$	741.622	0.0312401
$827$	− 34031.0i	− 1.43092i	−0.698651	−	0.715462i	$-0.746216\pi$
$827$	− 34031.0i	− 1.43092i	0.698651	−	0.715462i	$-0.253784\pi$
$828$	2139.71i	0.0898067i
$829$	−4931.55	−0.206610	−0.103305	−	0.994650i	$-0.532942\pi$
$829$	−4931.55	−0.206610	−0.103305	+	0.994650i	$0.532942\pi$
$830$	0	0
$831$	−4501.92	−0.187930
$832$	− 2902.15i	− 0.120930i
$833$	13748.0i	0.571836i
$834$	−690.138	−0.0286541
$835$	0	0
$836$	824.830	0.0341236
$837$	2211.60i	0.0913312i
$838$	− 25373.0i	− 1.04594i
$839$	38189.8	1.57146	0.785731	−	0.618568i	$-0.212287\pi$
$839$	38189.8	1.57146	0.785731	+	0.618568i	$0.212287\pi$
$840$	0	0
$841$	−23764.3	−0.974384
$842$	− 35915.6i	− 1.46999i
$843$	− 42106.8i	− 1.72033i
$844$	−57.5250	−0.00234608
$845$	0	0
$846$	−33683.7	−1.36888
$847$	− 371.687i	− 0.0150783i
$848$	20382.7i	0.825406i
$849$	−37488.0	−1.51541
$850$	0	0
$851$	−1463.59	−0.0589556
$852$	881.030i	0.0354268i
$853$	42966.8i	1.72469i	0.506325	+	0.862343i	$0.331003\pi$
$853$	42966.8i	1.72469i	−0.506325	+	0.862343i	$0.668997\pi$
$854$	−6199.37	−0.248405
$855$	0	0
$856$	−19406.8	−0.774898
$857$	17281.5i	0.688828i	0.938818	+	0.344414i	$0.111922\pi$
$857$	17281.5i	0.688828i	−0.938818	+	0.344414i	$0.888078\pi$
$858$	1276.85i	0.0508052i
$859$	−9316.75	−0.370062	−0.185031	−	0.982733i	$-0.559239\pi$
$859$	−9316.75	−0.370062	−0.185031	+	0.982733i	$0.559239\pi$
$860$	0	0
$861$	−6358.10	−0.251665
$862$	− 13413.5i	− 0.530005i
$863$	− 9647.65i	− 0.380544i	−0.981731	−	0.190272i	$-0.939063\pi$
$863$	− 9647.65i	− 0.380544i	0.981731	−	0.190272i	$-0.0609370\pi$
$864$	1699.93	0.0669361
$865$	0	0
$866$	−32083.3	−1.25893
$867$	− 25483.6i	− 0.998232i
$868$	− 51.8501i	− 0.00202754i
$869$	14236.5	0.555742
$870$	0	0
$871$	1834.70	0.0713735
$872$	24352.6i	0.945737i
$873$	48277.6i	1.87165i
$874$	42568.0	1.64746
$875$	0	0
$876$	−4293.75	−0.165608
$877$	− 19728.7i	− 0.759624i	−0.925064	−	0.379812i	$-0.875989\pi$
$877$	− 19728.7i	− 0.759624i	0.925064	−	0.379812i	$-0.124011\pi$
$878$	− 32304.3i	− 1.24171i
$879$	18464.1	0.708509
$880$	0	0
$881$	19473.9	0.744712	0.372356	−	0.928090i	$-0.378550\pi$
$881$	19473.9	0.744712	0.372356	+	0.928090i	$0.378550\pi$
$882$	32676.4i	1.24748i
$883$	49092.4i	1.87100i	0.353329	+	0.935499i	$0.385050\pi$
$883$	49092.4i	1.87100i	−0.353329	+	0.935499i	$0.614950\pi$
$884$	−118.365	−0.00450346
$885$	0	0
$886$	27599.5	1.04653
$887$	− 9292.86i	− 0.351774i	−0.984410	−	0.175887i	$-0.943721\pi$
$887$	− 9292.86i	− 0.351774i	0.984410	−	0.175887i	$-0.0562793\pi$
$888$	2430.12i	0.0918348i
$889$	4047.41	0.152695
$890$	0	0
$891$	4525.85	0.170170
$892$	− 2108.05i	− 0.0791288i
$893$	− 48112.0i	− 1.80292i
$894$	−52598.5	−1.96774
$895$	0	0
$896$	3949.89	0.147273
$897$	− 4731.10i	− 0.176106i
$898$	− 943.252i	− 0.0350520i
$899$	787.273	0.0292069
$900$	0	0
$901$	14136.7	0.522709
$902$	7845.88i	0.289622i
$903$	1405.53i	0.0517974i
$904$	6881.46	0.253179
$905$	0	0
$906$	55811.5	2.04659
$907$	− 37688.7i	− 1.37975i	−0.723928	−	0.689875i	$-0.757665\pi$
$907$	− 37688.7i	− 1.37975i	0.723928	−	0.689875i	$-0.242335\pi$
$908$	949.559i	0.0347051i
$909$	5789.42	0.211246
$910$	0	0
$911$	33049.6	1.20196	0.600979	−	0.799265i	$-0.294778\pi$
$911$	33049.6	1.20196	0.600979	+	0.799265i	$0.294778\pi$
$912$	− 65923.1i	− 2.39357i
$913$	− 4860.31i	− 0.176180i
$914$	28870.0	1.04479
$915$	0	0
$916$	−1026.44	−0.0370247
$917$	4917.06i	0.177073i
$918$	7906.43i	0.284260i
$919$	23148.0	0.830883	0.415442	−	0.909620i	$-0.363627\pi$
$919$	23148.0	0.830883	0.415442	+	0.909620i	$0.363627\pi$
$920$	0	0
$921$	13304.1	0.475986
$922$	− 12933.4i	− 0.461973i
$923$	− 1111.26i	− 0.0396290i
$924$	143.563	0.00511134
$925$	0	0
$926$	9374.21	0.332673
$927$	1246.34i	0.0441588i
$928$	− 605.131i	− 0.0214056i
$929$	23177.9	0.818561	0.409280	−	0.912409i	$-0.365780\pi$
$929$	23177.9	0.818561	0.409280	+	0.912409i	$0.365780\pi$
$930$	0	0
$931$	−46673.3	−1.64302
$932$	2355.98i	0.0828033i
$933$	− 28325.1i	− 0.993916i
$934$	−13979.8	−0.489757
$935$	0	0
$936$	−4481.13	−0.156485
$937$	34574.7i	1.20545i	0.797950	+	0.602724i	$0.205918\pi$
$937$	34574.7i	1.20545i	−0.797950	+	0.602724i	$0.794082\pi$
$938$	2873.18i	0.100014i
$939$	56959.1	1.97954
$940$	0	0
$941$	41831.2	1.44916	0.724578	−	0.689192i	$-0.242034\pi$
$941$	41831.2	1.44916	0.724578	+	0.689192i	$0.242034\pi$
$942$	53629.6i	1.85493i
$943$	− 29071.3i	− 1.00392i
$944$	5251.40	0.181058
$945$	0	0
$946$	1734.42	0.0596097
$947$	− 27231.2i	− 0.934419i	−0.884147	−	0.467209i	$-0.845259\pi$
$947$	− 27231.2i	− 0.934419i	0.884147	−	0.467209i	$-0.154741\pi$
$948$	5498.79i	0.188389i
$949$	5415.79	0.185252
$950$	0	0
$951$	124.534	0.00424635
$952$	− 2952.50i	− 0.100516i
$953$	40939.4i	1.39156i	0.718254	+	0.695781i	$0.244942\pi$
$953$	40939.4i	1.39156i	−0.718254	+	0.695781i	$0.755058\pi$
$954$	33600.3	1.14030
$955$	0	0
$956$	2188.87	0.0740515
$957$	2179.81i	0.0736292i
$958$	31601.3i	1.06575i
$959$	−4950.74	−0.166703
$960$	0	0
$961$	−28798.9	−0.966698
$962$	− 192.436i	− 0.00644945i
$963$	29839.0i	0.998491i
$964$	2094.60	0.0699820
$965$	0	0
$966$	7409.03	0.246772
$967$	46173.1i	1.53550i	0.640750	+	0.767750i	$0.278624\pi$
$967$	46173.1i	1.53550i	−0.640750	+	0.767750i	$0.721376\pi$
$968$	− 2821.78i	− 0.0936937i
$969$	−45721.8	−1.51579
$970$	0	0
$971$	−5153.91	−0.170337	−0.0851683	−	0.996367i	$-0.527143\pi$
$971$	−5153.91	−0.170337	−0.0851683	+	0.996367i	$0.527143\pi$
$972$	2764.06i	0.0912111i
$973$	− 97.8734i	− 0.00322474i
$974$	50069.0	1.64714
$975$	0	0
$976$	−43897.6	−1.43968
$977$	− 9692.13i	− 0.317378i	−0.987329	−	0.158689i	$-0.949273\pi$
$977$	− 9692.13i	− 0.317378i	0.987329	−	0.158689i	$-0.0507268\pi$
$978$	59026.5i	1.92992i
$979$	−16380.2	−0.534744
$980$	0	0
$981$	37443.3	1.21863
$982$	20811.6i	0.676299i
$983$	32915.7i	1.06800i	0.845483	+	0.534002i	$0.179313\pi$
$983$	32915.7i	1.06800i	−0.845483	+	0.534002i	$0.820687\pi$
$984$	−48269.5	−1.56380
$985$	0	0
$986$	2814.48	0.0909041
$987$	− 8373.97i	− 0.270057i
$988$	− 401.841i	− 0.0129395i
$989$	−6426.54	−0.206625
$990$	0	0
$991$	29477.9	0.944901	0.472451	−	0.881357i	$-0.343370\pi$
$991$	29477.9	0.944901	0.472451	+	0.881357i	$0.343370\pi$
$992$	− 762.560i	− 0.0244066i
$993$	10456.9i	0.334180i
$994$	1740.26	0.0555310
$995$	0	0
$996$	1877.28	0.0597227
$997$	31944.4i	1.01473i	0.861731	+	0.507366i	$0.169381\pi$
$997$	31944.4i	1.01473i	−0.861731	+	0.507366i	$0.830619\pi$
$998$	35268.4i	1.11864i
$999$	922.883	0.0292279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.4.b.c.199.1		4
5.2	odd	4		11.4.a.a.1.2	✓	2
5.3	odd	4		275.4.a.b.1.1		2
5.4	even	2	inner	275.4.b.c.199.4		4
15.2	even	4		99.4.a.c.1.1		2
15.8	even	4		2475.4.a.q.1.2		2
20.7	even	4		176.4.a.i.1.2		2
35.27	even	4		539.4.a.e.1.2		2
40.27	even	4		704.4.a.n.1.1		2
40.37	odd	4		704.4.a.p.1.2		2
55.2	even	20		121.4.c.f.81.2		8
55.7	even	20		121.4.c.f.27.1		8
55.17	even	20		121.4.c.f.3.2		8
55.27	odd	20		121.4.c.c.3.1		8
55.32	even	4		121.4.a.c.1.1		2
55.37	odd	20		121.4.c.c.27.2		8
55.42	odd	20		121.4.c.c.81.1		8
55.47	odd	20		121.4.c.c.9.2		8
55.52	even	20		121.4.c.f.9.1		8
60.47	odd	4		1584.4.a.bc.1.1		2
65.12	odd	4		1859.4.a.a.1.1		2
165.32	odd	4		1089.4.a.v.1.2		2
220.87	odd	4		1936.4.a.w.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.2	✓	2	5.2	odd	4
99.4.a.c.1.1		2	15.2	even	4
121.4.a.c.1.1		2	55.32	even	4
121.4.c.c.3.1		8	55.27	odd	20
121.4.c.c.9.2		8	55.47	odd	20
121.4.c.c.27.2		8	55.37	odd	20
121.4.c.c.81.1		8	55.42	odd	20
121.4.c.f.3.2		8	55.17	even	20
121.4.c.f.9.1		8	55.52	even	20
121.4.c.f.27.1		8	55.7	even	20
121.4.c.f.81.2		8	55.2	even	20
176.4.a.i.1.2		2	20.7	even	4
275.4.a.b.1.1		2	5.3	odd	4
275.4.b.c.199.1		4	1.1	even	1	trivial
275.4.b.c.199.4		4	5.4	even	2	inner
539.4.a.e.1.2		2	35.27	even	4
704.4.a.n.1.1		2	40.27	even	4
704.4.a.p.1.2		2	40.37	odd	4
1089.4.a.v.1.2		2	165.32	odd	4
1584.4.a.bc.1.1		2	60.47	odd	4
1859.4.a.a.1.1		2	65.12	odd	4
1936.4.a.w.1.2		2	220.87	odd	4
2475.4.a.q.1.2		2	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	275.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.73205i q^{2} -7.92820i q^{3} +0.535898 q^{4} -21.6603 q^{6} -3.07180i q^{7} -23.3205i q^{8} -35.8564 q^{9} +O(q^{10})\)	\(q-2.73205i q^{2} -7.92820i q^{3} +0.535898 q^{4} -21.6603 q^{6} -3.07180i q^{7} -23.3205i q^{8} -35.8564 q^{9} -11.0000 q^{11} -4.24871i q^{12} +5.35898i q^{13} -8.39230 q^{14} -59.4256 q^{16} +41.2154i q^{17} +97.9615i q^{18} -139.923 q^{19} -24.3538 q^{21} +30.0526i q^{22} -111.354i q^{23} -184.890 q^{24} +14.6410 q^{26} +70.2154i q^{27} -1.64617i q^{28} +24.9948 q^{29} +31.4974 q^{31} -24.2102i q^{32} +87.2102i q^{33} +112.603 q^{34} -19.2154 q^{36} -13.1436i q^{37} +382.277i q^{38} +42.4871 q^{39} +261.072 q^{41} +66.5359i q^{42} -57.7128i q^{43} -5.89488 q^{44} -304.224 q^{46} +343.846i q^{47} +471.138i q^{48} +333.564 q^{49} +326.764 q^{51} +2.87187i q^{52} -342.995i q^{53} +191.832 q^{54} -71.6359 q^{56} +1109.34i q^{57} -68.2872i q^{58} -88.3693 q^{59} +738.697 q^{61} -86.0526i q^{62} +110.144i q^{63} -541.549 q^{64} +238.263 q^{66} -342.359i q^{67} +22.0873i q^{68} -882.836 q^{69} -207.364 q^{71} +836.190i q^{72} -1010.60i q^{73} -35.9090 q^{74} -74.9845 q^{76} +33.7898i q^{77} -116.077i q^{78} -1294.23 q^{79} -411.441 q^{81} -713.261i q^{82} +441.846i q^{83} -13.0512 q^{84} -157.674 q^{86} -198.164i q^{87} +256.526i q^{88} +1489.11 q^{89} +16.4617 q^{91} -59.6743i q^{92} -249.718i q^{93} +939.405 q^{94} -191.944 q^{96} -1346.42i q^{97} -911.314i q^{98} +394.420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{4} - 52 q^{6} - 88 q^{9}+O(q^{10}) \) 4 * q + 16 * q^4 - 52 * q^6 - 88 * q^9	\( 4 q + 16 q^{4} - 52 q^{6} - 88 q^{9} - 44 q^{11} + 8 q^{14} - 16 q^{16} - 144 q^{19} + 152 q^{21} - 504 q^{24} - 80 q^{26} - 288 q^{29} - 68 q^{31} + 104 q^{34} - 160 q^{36} - 800 q^{39} + 1072 q^{41} - 176 q^{44} - 628 q^{46} + 780 q^{49} - 328 q^{51} + 220 q^{54} + 240 q^{56} - 1268 q^{59} + 1680 q^{61} - 448 q^{64} + 572 q^{66} - 1924 q^{69} - 1356 q^{71} - 12 q^{74} + 864 q^{76} - 632 q^{79} - 2588 q^{81} + 1472 q^{84} - 312 q^{86} + 3684 q^{89} + 2560 q^{91} + 1984 q^{94} - 1904 q^{96} + 968 q^{99}+O(q^{100}) \) 4 * q + 16 * q^4 - 52 * q^6 - 88 * q^9 - 44 * q^11 + 8 * q^14 - 16 * q^16 - 144 * q^19 + 152 * q^21 - 504 * q^24 - 80 * q^26 - 288 * q^29 - 68 * q^31 + 104 * q^34 - 160 * q^36 - 800 * q^39 + 1072 * q^41 - 176 * q^44 - 628 * q^46 + 780 * q^49 - 328 * q^51 + 220 * q^54 + 240 * q^56 - 1268 * q^59 + 1680 * q^61 - 448 * q^64 + 572 * q^66 - 1924 * q^69 - 1356 * q^71 - 12 * q^74 + 864 * q^76 - 632 * q^79 - 2588 * q^81 + 1472 * q^84 - 312 * q^86 + 3684 * q^89 + 2560 * q^91 + 1984 * q^94 - 1904 * q^96 + 968 * q^99

Introduction

L-functions

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